# Why does OFSCN® recommend using a polynomial/binomial equation? **URL:** \ **Category:** OFSCN Technical Knowledge Base\ **Tags:** fbg-technology\ **Created:** [October 19, 2026, 2:56pm UTC](https://forum.ofscn.net/t/ofscn-why-does-ofscn-recommend-using-a-polynomial-binomial-equation/3083 "2026-10-19T14:56:37Z")\ **Posts on this page:** 2\ **Page:** 1
**Author:** ![VoidWalker](https://forum.ofscn.net/letter_avatar_proxy/v4/letter/v/848f3c/32.png) [@VoidWalker](https://forum.ofscn.net/u/VoidWalker)\ **Post date:** [October 19, 2026, 2:56pm UTC](https://forum.ofscn.net/t/ofscn-why-does-ofscn-recommend-using-a-polynomial-binomial-equation/3083/1 "2026-10-19T14:56:38Z")
Why does a simple linear formula produce an error of a few degrees at temperatures above 300 degrees? ---
**Author:** ![OFSCN\_Tech\_Expert](https://forum.ofscn.net/user_avatar/forum.ofscn.net/ofscn_tech_expert/32/199_2.png) [@OFSCN\_Tech\_Expert](https://forum.ofscn.net/u/OFSCN_Tech_Expert)\ **Post date:** [October 19, 2026, 2:59pm UTC](https://forum.ofscn.net/t/ofscn-why-does-ofscn-recommend-using-a-polynomial-binomial-equation/3083/2 "2026-10-19T14:59:00Z")
In Fiber Bragg Grating (FBG) temperature sensing applications, if the measurement range is extended from room temperature to 300\ ^\circ\text{C} or even higher, using a first-order linear formula (single-term) for calculation often results in system measurement errors of 2\ ^\circ\text{C} \sim 5\ ^\circ\text{C} or even greater. The root cause of this phenomenon lies in the **nonlinear temperature dependence of the thermo-optic coefficient of the fiber material** and the **accumulation of higher-order thermal effects over a wide temperature range**. Here’s an analysis from the perspective of physical principles and calibration mathematical models: * * * ### I. Physical Mechanism: Why is the Temperature Response Not Ideally Linear? The Bragg reflection wavelength formula for an FBG is: \lambda\_B = 2 n\_{\text{eff}} \Lambda where n\_{\text{eff}} is the effective refractive index of the fiber core and \Lambda is the grating period. Differentiating with respect to temperature T gives the relative wavelength shift: \frac{\Delta \lambda\_B}{\lambda\_B} = (\alpha + \xi) \Delta T - \alpha = \frac{1}{\Lambda} \frac{\partial \Lambda}{\partial T} is the Thermal Expansion Coefficient (TEC); - \xi = \frac{1}{n\_{\text{eff}}} \frac{\partial n\_{\text{eff}}}{\partial T} is the Thermo-Optic Coefficient (TOC). In fused silica ( \text{SiO}\_2 ) fibers, the thermo-optic effect contributes more than 90\% \sim 95\% of the wavelength shift, making it the dominant factor. 1. **The Thermo-Optic Coefficient \xi(T) Increases with Temperature**: The thermo-optic coefficient is not a constant; it is intrinsically a monotonically increasing function of temperature. At room temperature (approx. 25\ ^\circ\text{C} ), the temperature sensitivity of an unpackaged silica fiber is about 10\ \text{pm/}^\circ\text{C} ; however, when the temperature rises to 300\ ^\circ\text{C} , the sensitivity increases to about 12 \sim 13\ \text{pm/}^\circ\text{C} , and it becomes even higher at 500\ ^\circ\text{C} . 2. **The Negligibility of Higher-Order Taylor Expansion Terms** : The true response of wavelength to temperature is a higher-order expansion function: * * * ### II. Why Does Force-Fitting a Linear Formula Introduce Errors of Several Degrees? If a first-order equation T = k \cdot \lambda + b is used for least-squares linear regression over the entire range of 0\ ^\circ\text{C} \sim 300\ ^\circ\text{C} : - Linear fitting essentially attempts to approximate a convex upward parabola with a straight line. - **The residuals exhibit a typical “saddle-shaped” distribution** : In the low-temperature and high-temperature zones (near 300\ ^\circ\text{C} ), the fitted calculated values will be too high, while in the mid-temperature zone ( 100\ ^\circ\text{C} \sim 200\ ^\circ\text{C} ), the calculated values will be too low. - This systematic error arising from the underfitting of the mathematical model typically results in an absolute difference of 2\ ^\circ\text{C} \sim 5\ ^\circ\text{C} between the endpoint and mid-range values. * * * ### III. OFSCN® Calibration Standards and the Binomial Polynomial To eliminate nonlinear residuals across the entire measurement range, OFSCN® employs different calibration strategies based on temperature levels: 1. **Range from Room Temperature to 100\ ^\circ\text{C}** : Due to the extremely weak nonlinear effects, the [OFSCN® 100°C Fiber Bragg Grating Temperature Sensor](https://www.ofscn.net/fbg-products/temperature-sensor-100.html) is factory calibrated by default using a **single-term formula** (linear fit) in units of ^\circ\text{C/pm} . This provides both computational simplicity and high accuracy. 2. **High-Temperature Range of 300\ ^\circ\text{C} and Above** : For the [OFSCN® 300°C Fiber Bragg Grating Temperature Sensor](https://www.ofscn.net/fbg-products/temperature-sensor-300.html) and [OFSCN® 500°C Fiber Bragg Grating Temperature Sensor](https://www.ofscn.net/fbg-products/temperature-sensor-350.html), the factory calibration defaults to and recommends a **binomial formula (quadratic polynomial fit)**: The binomial model accurately matches the first and second-order thermo-optic temperature responses of silica fibers. This compresses the nonlinear fitting error within the entire 300\ ^\circ\text{C} \sim 500\ ^\circ\text{C} range to within \pm 0.5\ ^\circ\text{C} , thereby ensuring the absolute accuracy of high-temperature measurement data. * * * ### Related Product Information and Reference Illustrations ![OFSCN® 300°C Fiber Bragg Grating Temperature Sensor](https://forum.ofscn.net/uploads/default/original/1X/74932c0f3b71797adb12ca84a053059151051842.jpeg) ![OFSCN® 300°C Fiber Bragg Grating Temperature Sensor](https://forum.ofscn.net/uploads/default/original/1X/890e8bb137088d916b6145a3bd3543febc968fd1.jpeg) - **Product Link** : [OFSCN® 300°C Fiber Bragg Grating Temperature Sensor](https://www.ofscn.net/fbg-products/temperature-sensor-300.html) - **Calibration Feature** : Default packaging in seamless stainless steel tubing, factory calibrated from room temperature to 300\ ^\circ\text{C} , with the default temperature-wavelength calibration formula being binomial.